Let Y_T=(Y(t), t in [0,T]) be a real ergodic diffusion process which drift depends on an unkown parameter qo. Our aim is to estimate qo from a discrete observation of the process Y_T, (Y(kd),k=0,...n), for a fixed and small d, as T=nd goes to infinity. For that purpose, we adapt the Generalized Method of Moments to the anticipative and approximate discrete-time trapezoidal scheme, and then to Simpson's. Under some general assumptions, the trapezoidal scheme (respectively Simpson's scheme) provides an estimation of qo with a bias of order d **2 (resp. d **4). Moreover, this estimator is asymptotically normal. These results generalize Bergstrom's, which were obtained for a Gaussian diffusion process, which drift is linear in q.